According to the summary
Although the work doesn't provide a definitive answer, it rules out one possible explanation. [R]esearchers point to three different factors that could independently or in combination have rid Brown’s body of HIV.
Unfortunately the summary forgets to mention the explanation that was ruled out or even clearly delineate the three different factors (though the latter was more the fault of the original article).
From my reading of TFA:
Explanation 1: Conditioning: The radiation that destroyed his immune system also killed off the HIV (because HIV lives in the cells of the immune system).
Explanation 2: Shiny new immune system: The new bone marrow had a mutation that was immune to HIV and that cured him (maybe by detecting and killing HIV infected cells?).
Explanation 3: Graft vs host: The new immune system killed off his old one, not just the HIV infected cells but all the old immune systems cells including those infected with HIV.
So the researchers took chimps, extracted some stem cells (bone marrow?), infected them with SHIV (Simian HIV), destroyed their immune systems with radiation, then reinjected the uninfected stem cells.
The SHIV quickly came back which rules out explanation 1.